segmented circle

The diagram above shows a circular plane, centered at the origin 'O', has a radius $7 cm$. Two identical rectangular strips, each having width $2 cm$, are thoroughly cut off from the circular plane along x & y axes. Thus four identical segments of the circular plane are left over.

Question: How to calculate the solid angle subtended by the remaining plane (consisting of four identical segments) at a point $P (0, 0, 4cm)$ lying on the z-axis (normally outwards to the plane of paper)?

Solid angle $(\omega)$ subtended by the circular plane, with a radius $r$ at any point lying on the geometrical axis at a distance $h$ from the center, is given as $$\omega=2\pi \left(1-\frac{h}{\sqrt{h^2+r^2}}\right)$$ And the solid angle $(\omega)$ subtended by any rectangle of size $l*b$ at any point lying on the perpendicular axis at a distance $d$ from the center is given by standard formula as $$\omega=4sin^{-1}\left(\frac{lb}{\sqrt{(l^2+4d^2)(b^2+4d^2)}}\right)$$

But how to apply these formula on the non radial segments of a circle? Any help is greatly appreciated.


The distance d=4 cm, radius R=7cm and gap g=2cm give a solid angle of 1.70538588 if we use my formula (C6) in Solid angle of a Rectangular Plate.

  • 1
    $\begingroup$ The solution seems to be very lengthy, don't you have any shortcut method to deal with such problems? $\endgroup$ May 17 '15 at 19:28
  • $\begingroup$ Whilst this may theoretically answer the question, it would be preferable to include the essential parts of the answer here, and provide the link for reference. $\endgroup$
    – user642796
    May 18 '15 at 7:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.